The Simplest Layout Spaces of the Graphical-Analytical Method for Flow Path Variation
Abstract Generated abstract
This paper develops ready-made calculation elements for applying the graph-analytical method of flow path variation to stationary conducting fields with prescribed electrode potentials. It derives conductance expressions for sectors of spherical and cylindrical capacitor fields, introducing expansion coefficients that relate each radial-flow conductance to the corresponding plane-capacitor conductance. The authors then propose using these standard fields to approximate parts of more general radial-like fields, with error estimates based on variation in field-line lengths and curvature. As an example, they outline the calculation of conductance per unit length for centered elliptical cylinders by iteratively varying tube boundaries to maximize total conductance, including a numerical formulation and a simplified analytical approximation for limited relative surface expansion.
Full Text
SIMPLEST LAYOUT SPACES OF THE GRAPH-ANALYTICAL METHOD OF FLOW PATH VARIATION
P. P. Galinskii, V. S. Sokolov
(Presented by the Scientific Seminar on the Theoretical Foundations of Electrical Engineering and by the Laboratory of Rotary ESG, Research Institute of Nuclear Physics)
To make convenient the practical use of flow path variations [1] in the graph-analytical method for calculating conductances or capacitances, it is necessary to have a set of ready-made solutions for frequently encountered configurations of the spaces occupied by individual parts of the flow.
In what follows, a stationary field in a conducting medium with prescribed constant electrode potentials will be meant.
Partial conductances are most simply expressed for the model fields of spherical and cylindrical capacitors.
Fig. 1
Let us isolate a sector of a spherical capacitor with central solid angle \(\omega\) and with radii of the inner and outer plates \(r_1\) and \(r_2\) (Fig. 1).
The differential resistance on an element \(dx\) of the length of the flow is equal to
\[ dR=\frac{dx}{\gamma S(x)}=\frac{dx}{\gamma \omega x^2}, \tag{1} \]
where \(x\) is the variable radius,
\(\gamma\) is the specific conductivity.
Integrating expression (1) within the limits from \(r_1\) to \(r_2\), we obtain the resistance along the flow path, from which it is not difficult to pass to the conductance
\[ G=\frac{\gamma \omega r_1 r_2}{r_2-r_1} =\frac{\gamma \omega r_1^2}{l}\cdot\frac{r_2}{r_1} =\frac{\gamma S_1}{l}\cdot\sqrt{\frac{S_2}{S_1}}, \tag{2} \]
where \(l=r_2-r_1\) is the length of the field line between the surfaces of the beginning of the flow \(S_1=\omega r_1^2\) and of the end of the flow \(S_2=\omega r_2^2\). Here and below, index 1 refers to the smaller surface, and index 2 to the larger one.
Introduce new notation:
\[ G_0=\frac{\gamma S_1}{l} \]
—the conductance of a uniform (non-expanding) field corresponding to a plane capacitor with plate area \(S_1\) and the same length of lines of force \(l\);
\[ r_2^*=\frac{r_2}{r_1} \]
—the relative radius;
\[ S_2^*=\frac{S_2}{S_1} \]
—the relative surface, characterizing the degree of expansion of the flux, and write the conductance (2) for the sector of a spherical capacitor in another form:
\[ G=G_0 r_2^* = G_0 \sqrt{S_2^*}=G_0 K_{\mathrm{s}}. \tag{3} \]
\(K_{\mathrm{s}}\) is a coefficient taking into account the expansion of the flux in a spherical capacitor,
\[ K_{\mathrm{s}}=\sqrt{S_2^*}. \tag{4} \]
Performing analogous operations for the sector of a cylindrical capacitor (Fig. 2), we obtain the following expression for its conductance:
\[ G=\frac{\gamma \alpha h}{\ln \dfrac{r_2}{r_1}} =\frac{\gamma \alpha h r_1}{r_2-r_1}\cdot \frac{r_2-r_1}{r_1 \ln \dfrac{r_2}{r_1}} = \frac{\gamma S_1}{l}\cdot K_{\mathrm{c}}. \tag{5} \]
where \(S_1=\alpha r_1 h\) is the surface of the smaller electrode,
\(l=r_2-r_1\) is the length of a line of force,
\(\alpha\) is the central angle of the cylindrical sector,
\(h\) is the height of the cylinder,
\(K_{\mathrm{c}}\) is a coefficient taking into account the expansion of the flux in a cylindrical capacitor,
\[ K_{\mathrm{c}}= \frac{\dfrac{r_2}{r_1}-1}{\ln \dfrac{r_2}{r_1}} = \frac{r_2^*-1}{\ln r_2^*} = \frac{S_2^*-1}{\ln S_2^*}. \tag{6} \]
Fig. 2
For a cylinder, the relative radius and relative surface are equal to each other,
\[ r_2^*=\frac{r_2}{r_1} = \frac{\alpha h r_2}{\alpha h r_1} = \frac{S_2}{S_1} = S_2^*. \tag{7} \]
Finally, for the conductance of the sector of a cylindrical capacitor we have the expression
\[ G=G_0 K_{\mathrm{c}}, \tag{8} \]
where
\[ G_0=\frac{\gamma S_1}{l} \]
is the conductance of a plane capacitor.
Let us note that the values of the flux-expansion coefficients (4) and (6) approach unity if the radii of the electrodes become close to one another or both are large in comparison with the length of the lines of force.
Graphs of the flow-expansion coefficients as a function of the relative surface are given in Fig. 3.
Passing to radial fields, close in configuration to the above-considered standard fields of spherical and cylindrical capacitors, and dividing the flow into parts in accordance with [1], we shall calculate the conductances of these parts by formulas derived for the standard fields. In doing so, as the calculated length of a line of force we take the average for the given segment, and estimate the error by the ratio of the difference between the maximum and minimum lengths of the lines of force on the segment to the average:
\[ \delta=\frac{l_{\max}-l_{\min}}{l_{\mathrm{av}}}\cdot 100\%. \tag{9} \]
One should also take into account the deviation of the lines of force from straight lines, either by the ratio of the maximum distance between the straight line and the curve to the length of the straight line, or by the relative difference of their lengths.
As an example, let us determine the conductance per unit length of centered elliptical cylinders (Fig. 4). By virtue of symmetry we consider one fourth of the cross section.
The following sequence of operations may be indicated:
- A sketch construction of the field pattern in accordance with the known principles [2]. In this case we divide the total flow into tubes so that, for each tube, condition (9) of closeness to a radial field proves acceptable.
Fig. 3. Graphs of the conductance-expansion coefficients
\(K_{\mathrm{ц}}=(s-1)\ln^{-1}s,\ K_{\mathrm{c}}=s^{0.5}\)
Fig. 4
-
Selection of a pair of neighboring tubes numbered 1 and 2, located near the vertical line \(a_0b_0\) (Fig. 4), and variation of the position of the end of the boundary line (for example, in Fig. 4: \(b_1^1,\ b_1^2,\ b_1^3\)) with the beginning of this line fixed (\(a_1\)), in order to determine the true position of the end of the boundary line from the maximum total conductance of these two tubes.
-
Consideration of the 2nd and 3rd tubes and repetition of the described operations for this pair of tubes, and so on.
-
Repetition of the process in the opposite direction from the other initial line of force (the horizontal \(a_{00}b_{00}\) in Fig. 4).
-
The position of the boundary of the meeting line is taken as the average of those obtained when moving from opposite sides. Of course, the position of each boundary line depends on the position of all the others. Therefore corrections of all already established boundaries are possible at each subsequent step. But in practice such corrections are expedient only when...
significant corrections for the nearest neighboring boundary as well.
- After the boundaries of the tubes have been established, the conductances of the tubes and the total conductance are calculated.
For illustration, let us describe the calculation for one pair of tubes numbered \(a\) and \(b\). Let \(S_1\) and \(S_2\) be the total surfaces, for these tubes, of the beginning and the end of the flow of the current-density vector (electrodes):
\[ S_1 = S_{1a} + S_{1b}, \]
\[ S_2 = S_{2a} + S_{2b}. \]
The lengths of the mean lines of force of the tubes are \(l_a\) and \(l_b\). The field of each tube will be regarded as close to a radial cylindrical field. Then the total conductance of the tubes is
\[ G_{ab} = G_a + G_b = G_{0a}K_a + G_{0b}K_b = \]
\[ = \frac{\gamma S_{1a}}{l_a}\cdot \frac{S_a^* - 1}{\ln S_a^*} + \frac{\gamma S_{1b}}{l_b}\cdot \frac{S_b^* - 1}{\ln S_b^*}, \]
where
\[ S_a^* = \frac{S_{2a}}{S_{1a}}, \qquad G_{0a} = \frac{\gamma S_{1a}}{l_a}. \]
The quantities \(S_b^*\) and \(G_{0b}\) are determined analogously.
The expression for the total conductance of the tubes can be represented in the form
\[ G_{ab} = G_{0a}\left(K_a + \frac{G_{0b}}{G_{0a}}K_b\right) = G_{0a}\left(K_a + \frac{S_{1b}}{S_{1a}}\cdot \frac{l_a}{l_b}\cdot K_b\right). \]
The initial surface can always be divided into equal parts, that is, \(S_{1a}=S_{1b}=\ldots S_{1k}\); then the multiplier before \(K_b\) in the last expression will be equal to the ratio of the lengths of the mean lines of force of the sections, \(l_a/l_b=l_a^*\). But for small variations in the position of the end of the boundary line of force, the magnitude of the mean lines of force of the tubes changes insignificantly and may be regarded as constant. Then finally one should find the maximum of the total conductance as a function of \(S_a^*\):
\[ G_{ab} = G_{0a}(K_a + l_a^*K_b) = \frac{\gamma S_{1a}}{l_a}\left[ \frac{S_a^* - 1}{\ln S_a^*} + l_a^*\cdot \frac{(\eta - S_a^*) - 1}{\ln(\eta - S_a^*)} \right], \tag{10} \]
where
\[ \eta = \frac{2S_2}{S_1} \]
is a constant quantity.
Analytical determination of the maximum of this function leads to the solution of a transcendental equation; therefore, in the general case it is more convenient to find the maximum by numerical enumeration. However, if the relative surface \(S^*\) does not exceed two, then, in accordance with the graph (Fig. 3), for the space-expansion coefficients expression (6) may approximately be replaced by (4), that is, one may put \(K=1/S^*\). Then the value of \(S_a^*\) corresponding to the maximum of the total conductance of the two tubes is found analytically in an elementary manner:
\[ S_a^* = \frac{\eta}{(l_a^*)^2 + 1} = \frac{2S_2}{S_1\left[(l_a/l_b)^2 + 1\right]}. \tag{11} \]
Thus the actual position of the boundary line is found for the prescribed positions of the extreme lines of force.
LITERATURE
-
P. P. Galinskii. On one technique for dividing the flow in the method of expected flow paths. Izv. TPI, vol. 210 (in press).
-
V. A. Govorkov. Electric and Magnetic Fields. GEI. Moscow–Leningrad, 1960.